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Consider a compact connected surface $M$ of some genus $\gamma \geq 2$. A particular case of a famous result of Colin de Verdière (see Construction de laplaciens dont une partie finie du spectre est donnée) says that if we fix $\gamma$ and select a finite sequence $\lambda_0 = 0 < \lambda_1 < \dotsb <\lambda_n$ (note the strict inequality, which is meant to avoid potential constraints on multiplicity in dimension $2$), then we can find a smooth metric $g$ on $M$ such that the given $\lambda_i$'s are the first $n$ eigenvalues of the Laplacian associated to $g$.

Now, my question (which could be trivial, but unfortunately I have no way of knowing as I am totally ignorant of French and nearly as ignorant of the graph-theoretic methods that de Verdière uses) is, if we do not fix $\gamma$, could we also arrange that the metric $g$ be hyperbolic, that is, constant negative curvature? If not, could we at least say that we can pick a $g$ whose curvature is arbitrarily close to being constant?

The reason I even remotely hope for a positive answer here is due to a result of Lohkamp (see Curvature $h$-principles), which says that in compact connected manifolds of higher dimensions, one can improve de Verdière's result in that one can additionally constrain the metric $g$ to have an arbitrary negative upper bound on the Ricci curvature.

Any insights are highly appreciated, thanks!

Edit after Igor Rivin's comment: I should explicitly mention again that I do not want to fix the genus of the surface. In other words, given a finite increasing sequence (starting from $0$) as above, can we find a compact hyperbolic (or close to hyperbolic) surface $(M, g)$ of some genus $\gamma \geq 2$ such that the first $n$ eigenvalues of $(M, g)$ agree with the given sequence?

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    $\begingroup$ There are restrictions on the spectrum of a genus $g$ hyperbolic surface: projecteuclid.org/euclid.dmj/1253020546 $\endgroup$
    – Ian Agol
    Oct 20, 2015 at 23:55
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    $\begingroup$ A remark on the first question. The moduli space of hyperbolic structures on a surface is finite dimensional (6g-6). So it cannot easily (for instance differentiably) map onto a space with more than 6g-6 parameters. $\endgroup$
    – Thomas
    Oct 21, 2015 at 7:35
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    $\begingroup$ Actually, the question is much more interesting if you read it as not restricting the genus. Then both the comments become irrelevant (or at least less relevant) $\endgroup$
    – Igor Rivin
    Oct 21, 2015 at 10:17
  • $\begingroup$ @IgorRivin Thanks, that is actually what I wanted to ask, but got a bit confused and did not word it properly. Now I have edited it. $\endgroup$
    – SMS
    Oct 21, 2015 at 10:47

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Vedrin Šahović in his unpublished thesis [Approximations of Riemannian manifolds with linear curvature constraints, 2009] proved that any compact metric space can be appoximated by hyperbolic surfaces $M_n$ with constant curvature $\kappa_n$. In this case $\kappa_n\to -\infty$ and $\gamma_n\to \infty$.

His construction is quite flexible — it is possible to get in addition convergence of the first $n$ eigenvalues (but please, do not ask me how to do it). So at least one can get a positive answer to approximate version of your question — we can get a hyperbolic manifold with first $n$ eigenvalues arbitrarily close to the given sequence.

By the way, the same holds in all dimensions.

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